White House chief strategist Stephen Bannon said Wednesday he was removed from a permanent seat on the National Security Council because it was no longer in danger of being “operationalized” for non-security purposes, as he asserted was the case under former Obama national security adviser Susan E. Rice.

Mr. Bannon, who was taken off the NSC by President Trump, said new national security adviser H.R. McMaster “has returned the NSC to its proper function.”

“Susan Rice operationalized the NSC during the last administration,” Mr. Bannon said in a statement. “I was put on to ensure that it was de-operationalized.”

The president took Mr. Bannon off the NSC’s principals’ committee, the main policy-making group for national security, less than three months after his appointment to the post had raised criticism about potential politicization of the council.

Mr. Bannon’s statement implied that he had been put on the NSC to monitor Mr. Trump’s first national security adviser, Michael Flynn. The president fired Mr. Flynn in February after he acknowledged misleading Vice President Mike Pence about the extent of his contacts with Russia’s ambassador to the U.S., Sergey Kislyak, during the presidential transition.

Ms. Rice has resurfaced prominently this week to defend herself against accusations that she “unmasked” Trump officials who were mentioned in classified intelligence documents while she was working at the White House. She denied revealing the identities of Trump associates who were allegedly mentioned in conversations by Russian officials caught up in U.S. surveillance, and denied leaking any such information to the media.

In the White House staff reshuffling, Mr. Trump also downgraded the role of Homeland Security adviser Tom Bossert on the NSC, according to a presidential memorandum to be filed in the Federal Register. The moves were sought by Mr. McMaster.

Under those moves, Director of National Intelligence Dan Coats and Joint Chiefs of Staff Chairman Gen. Joseph Dunford are again “regular attendees” of the NSC’s principals committee.

Mr. Bannon’s inclusion in the principals committee at the start of Mr. Trump’s presidency drew criticism from some outside the White House as a potential for political interference on the NSC.

White House officials say Mr. Bannon will still be able to attend NSC meetings — he reportedly had attended only one — and that the move doesn’t change his status as one of the president’s most influential advisers.

While the White House portrayed Mr. Bannon’s shift as an effort to ensure de-politicization of national security, some Democrats said he should not have had a seat on the NSC in the first place.

“President Trump’s decision to add Steve Bannon to NSC put politics ahead of national security,” said House Democratic Whip Steny Hoyer of Maryland. “His removal was long overdue.”

Rep. Elijah E. Cummings of Maryland, the top Democrat on the House Oversight and Government Reform Committee, said the removal of Mr. Bannon from the NSC came a few days after committee Democrats asked the White House to reveal the names of all presidential staffers involved in dealings with House Intelligence Committee Chairman Devin Nunes, who visited the White House last month as part of his investigation into President Trump’s claims that the Obama administration conducted surveillance on Trump campaign officials.

“One key question is whether Mr. Bannon was involved in any way with the dealings between NSC staffers and Chairman Nunes or his trip to the White House,” Mr. Cummings said. “We asked the White House to tell us by this Friday whether National Security Advisor McMaster or White House Counsel [Don] McGahn was personally aware of these activities and whether anyone else at the White House was involved.”

He said Mr. Bannon’s post on the NSC was “a role he was wholly unqualified for.”

“We need to know what in the world is going on,” Mr. Cummings said.

Rep. Ileana Ros-Lehtinen, Florida Republican, said Mr. Bannon’s removal from the NSC is “welcome news.” She noted on Twitter that she called on Mr. Bannon in January to “write himself out” of the council.